134 research outputs found

    On the complexity of solving ordinary differential equations in terms of Puiseux series

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    We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given by Grigoriev [10] for Riccatti differential polynomials associated to ordinary linear differential operators. In this paper, we get the same bound for arbitrary differential polynomials. The algorithm is based on a differential version of the Newton-Puiseux procedure for algebraic equations

    An Algorithm for Computing the Limit Points of the Quasi-component of a Regular Chain

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    For a regular chain RR, we propose an algorithm which computes the (non-trivial) limit points of the quasi-component of RR, that is, the set W(R)ˉ∖W(R)\bar{W(R)} \setminus W(R). Our procedure relies on Puiseux series expansions and does not require to compute a system of generators of the saturated ideal of RR. We focus on the case where this saturated ideal has dimension one and we discuss extensions of this work in higher dimensions. We provide experimental results illustrating the benefits of our algorithms

    Analogue of Newton-Puiseux series for non-holonomic D-modules and factoring

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    We introduce a concept of a fractional-derivatives series and prove that any linear partial differential equation in two independent variables has a fractional-derivatives series solution with coefficients from a differentially closed field of zero characteristic. The obtained results are extended from a single equation to DD-modules having infinite-dimensional space of solutions (i. e. non-holonomic DD-modules). As applications we design algorithms for treating first-order factors of a linear partial differential operator, in particular for finding all (right or left) first-order factors

    Computing Puiseux series : a fast divide and conquer algorithm

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    Let F∈K[X,Y]F\in \mathbb{K}[X, Y ] be a polynomial of total degree DD defined over a perfect field K\mathbb{K} of characteristic zero or greater than DD. Assuming FF separable with respect to YY , we provide an algorithm that computes the singular parts of all Puiseux series of FF above X=0X = 0 in less than O~(Dδ)\tilde{\mathcal{O}}(D\delta) operations in K\mathbb{K}, where δ\delta is the valuation of the resultant of FF and its partial derivative with respect to YY. To this aim, we use a divide and conquer strategy and replace univariate factorization by dynamic evaluation. As a first main corollary, we compute the irreducible factors of FF in K[[X]][Y]\mathbb{K}[[X]][Y ] up to an arbitrary precision XNX^N with O~(D(δ+N))\tilde{\mathcal{O}}(D(\delta + N )) arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by FF with O~(D3)\tilde{\mathcal{O}}(D^3) arithmetic operations and, if K=Q\mathbb{K} = \mathbb{Q}, with O~((h+1)D3)\tilde{\mathcal{O}}((h+1)D^3) bit operations using a probabilistic algorithm, where hh is the logarithmic heigth of FF.Comment: 27 pages, 2 figure

    Singular Points of Real Quartic and Quintic Curves

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    There are thirteen types of singular points for irreducible real quartic curves and seventeen types of singular points for reducible real quartic curves. This classification is originally due to D. A. Gudkov. There are nine types of singular points for irreducible complex quartic curves and ten types of singular points for reducible complex quartic curves. There are 42 types of real singular points for irreducible real quintic curves and 49 types of real singular points for irreducible real quintic curves. The classification of real singular points for irreducible real quintic curves is originally due to Golubina and Tai. There are 28 types of singular points for irreducible complex quintic curves and 33 types of singular points for reducible complex quintic curves. We derive the complete classification with proof by using the computer algebra system Maple. We clarify that the classification is based on computing just enough of the Puiseux expansion to separate the branches. Thus, the proof consists of a sequence of large symbolic computations that cam be done nicely using Maple
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